Abstract
Let A be a central simple algebra with involution σ of the first or second kind. Let v be a valuation on the σ-fixed part F of Z(A). A σ-special v-gauge g on A is a kind of value function on A extending v on F, such that g(σ(x)x) = 2g(x) for all x in A. It is shown (under certain restrictions if the residue characteristic is 2) that if v is Henselian, then there is a σ-special v-gauge g if and only if σ is anisotropic, and g is unique. If v is not Henselian, it is shown that there is a σ-special v-gauge g if and only if σ remains anisotropic after scalar extension from F to the Henselization of F with respect to v; when this occurs, g is the unique σ-invariant v-gauge on A.
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J.-P. Tignol was partially supported by the F.R.S.-FNRS, Belgium. A.R. Wadsworth would like to thank J.-P. Tignol and UCL for their hospitality during several visits while this paper was developing.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Tignol, JP., Wadsworth, A.R. Valuations on algebras with involution. Math. Ann. 351, 109–148 (2011). https://doi.org/10.1007/s00208-010-0580-9
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DOI: https://doi.org/10.1007/s00208-010-0580-9